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Properties

Label 16.0.44336421273...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{4}\cdot 5^{8}\cdot 17^{4}$
Root discriminant $16.90$
Ramified primes $2, 3, 5, 17$
Class number $1$
Class group Trivial
Galois group $(C_2\times D_4):C_2^2$ (as 16T115)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -64, 128, -112, 32, -8, 58, -156, 177, -120, 72, -46, 34, -16, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 7*x^14 - 16*x^13 + 34*x^12 - 46*x^11 + 72*x^10 - 120*x^9 + 177*x^8 - 156*x^7 + 58*x^6 - 8*x^5 + 32*x^4 - 112*x^3 + 128*x^2 - 64*x + 16)
 
gp: K = bnfinit(x^16 - 2*x^15 + 7*x^14 - 16*x^13 + 34*x^12 - 46*x^11 + 72*x^10 - 120*x^9 + 177*x^8 - 156*x^7 + 58*x^6 - 8*x^5 + 32*x^4 - 112*x^3 + 128*x^2 - 64*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 7 x^{14} - 16 x^{13} + 34 x^{12} - 46 x^{11} + 72 x^{10} - 120 x^{9} + 177 x^{8} - 156 x^{7} + 58 x^{6} - 8 x^{5} + 32 x^{4} - 112 x^{3} + 128 x^{2} - 64 x + 16 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44336421273600000000=2^{24}\cdot 3^{4}\cdot 5^{8}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.90$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{28} a^{13} + \frac{1}{28} a^{12} - \frac{3}{28} a^{11} + \frac{5}{28} a^{10} - \frac{1}{14} a^{9} - \frac{2}{7} a^{8} + \frac{1}{14} a^{7} - \frac{3}{7} a^{6} + \frac{9}{28} a^{5} - \frac{13}{28} a^{4} + \frac{1}{14} a^{3} - \frac{3}{14} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{1736} a^{14} - \frac{2}{217} a^{13} + \frac{211}{1736} a^{12} - \frac{21}{124} a^{11} - \frac{159}{868} a^{10} - \frac{127}{868} a^{9} + \frac{17}{434} a^{8} - \frac{88}{217} a^{7} + \frac{241}{1736} a^{6} - \frac{181}{868} a^{5} - \frac{95}{868} a^{4} - \frac{19}{217} a^{3} + \frac{9}{31} a^{2} + \frac{73}{217} a + \frac{90}{217}$, $\frac{1}{881164088} a^{15} - \frac{87457}{440582044} a^{14} - \frac{9651827}{881164088} a^{13} + \frac{22667787}{220291022} a^{12} - \frac{19718504}{110145511} a^{11} + \frac{101873607}{440582044} a^{10} + \frac{11187487}{110145511} a^{9} + \frac{54445494}{110145511} a^{8} + \frac{308412953}{881164088} a^{7} - \frac{60166153}{220291022} a^{6} - \frac{14155882}{110145511} a^{5} - \frac{1416504}{110145511} a^{4} - \frac{9033134}{110145511} a^{3} + \frac{12307921}{31470146} a^{2} - \frac{54824110}{110145511} a - \frac{9491592}{110145511}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1200.99168735 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4):C_2^2$ (as 16T115):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 25 conjugacy class representatives for $(C_2\times D_4):C_2^2$
Character table for $(C_2\times D_4):C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.27200.2, \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.1088.2, 8.0.739840000.6, 8.4.23040000.1, 8.4.6658560000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$